The Chow \(t\)-structure on the \(\infty\)-category of motivic spectra. (English) Zbl 1489.14030

Bachmann, Tom; Kong, Hana Jia; Wang, Guozhen; Xu, Zhouli

Ann. Math. (2) 195, No. 2, 707-773 (2022).

The Morel-Voevodsky \(\mathbb{P}^1\)-stable \(\mathbb{A}^1\)-homotopy category \(\mathcal{SH}(k)\) of a field \(k\) offers a way to construct interesting (co)homology theories of algebraic varieties. Two theories introduced before its invention, algebraic \(K\)-theory and higher Chow groups (a.k.a. motivic cohomology) of smooth \(k\)-varieties, are representable in \(\mathcal{SH}(k)\), whereas algebraic cobordism is the prime example, due to Voevodsky, of a (co)homology theory created through its representing \(\mathbb{P}^1\)-spectrum \(\mathsf{MGL}\). It is built out of Thom spaces \(\mathrm{Th}(\gamma_k)\) of tautological bundles \(\gamma_k \to \mathrm{colim}_{n\to \infty} \mathrm{Gr}(k,n)\) on infinite Grassmann varieties and constitutes a close relative of the topological complex cobordism spectrum \(\mathrm{MU}\). Motivic stable (co)homotopy represented by the motivic sphere spectrum \(\mathbf{1}\in \mathcal{SH}(k)\) may be regarded as the “initial” example in the sense that all \(\mathbb{P}^1\)-spectra in \(\mathcal{SH}(k)\) are canonically modules over \(\mathbf{1}\).
One major technique used in the very difficult (at least in these four examples) task of computing (co)homology groups for a given theory is to build a filtration on its representing \(\mathbb{P}^1\)-spectrum. If its associated graded is more accessible, the filtration leads to a spectral sequence whose initial term may be computed using algebraic means, at least in a certain range. Often the way of building a filtration applies to every \(\mathbb{P}^1\)-spectrum naturally, thus leading to a filtration on the whole homotopy category \(\mathcal{SH}(k)\). Voevodsky’s slice filtration (which leads to “the” motivic spectral sequence for algebraic \(K\)-theory on smooth \(k\)-varieties) and Morel’s homotopy \(t\)-structure are two prominent examples.
This article introduces another filtration on \(\mathcal{SH}(k)\), the Chow \(t\)-structure. Its nonnegative part is the subcategory \(\mathcal{SH}(k)_{c\geq 0}\) generated under colimits and extensions by Thom spectra \(\mathrm{Th}(\xi)\) of virtual vector bundles \(\xi\in K(X)\) over smooth proper \(k\)-varieties. Theorem 3.14, the main result of this article, exhibits a close relation between the Chow \(t\)-structure and the algebraic cobordism spectrum \(\mathsf{MGL}\): If \(E\in \mathcal{SH}(k)_{c\geq 0}\), then the homotopy groups of its nonpositive truncation \(\tau_{c\leq 0}E = E_{c=0}\) can be canonically identified with Ext-groups of the \(\mathsf{MGL}\)-homology of \(E\) in the category of \(\mathrm{MU}_\ast\mathrm{MU}\)-comodules: \[ \pi_{2w-s,w}\tau_{c\leq 0}E \cong \mathrm{Ext}_{\mathrm{MU}_\ast\mathrm{MU}}^{s,2w}(\mathrm{MU}_\ast,\mathsf{MGL}_{2\ast,\ast}E) \] In this and the following results, the characteristic of \(k\) is implicitly inverted if it is positive. Taking \(E=\mathbf{1}\) the motivic sphere spectrum reproduces the \(E^2\)-term of the topological Adams-Novikov spectral sequence. Hence the Chow \(t\)-structure provides a vast generalization (to arbitrary base fields and coefficients) of work by Gheorghe, Wang, and Xu on the \(2\)-completed motivic sphere spectrum over the complex numbers [B. Gheorghe, Doc. Math. 23, 1077–1127 (2018; Zbl 1407.55007)], which in turn allowed amazing progress in classical homotopy theory: Isaksen, Wang, and Xu used it to extend the range in which the infamous stable homotopy groups of spheres are known by a factor of about \(\tfrac{3}{2}\). [D. C. Isaksen et al., Proc. Natl. Acad. Sci. USA 117, No. 40, 24757–24763 (2020; Zbl 1485.55017)].
One consequence of Theorem 3.14 reconstructs the motivic Adams spectral sequence (based on motivic cohomology with \(\mathbb{Z}/p\)-coefficients) for \(\mathbb{P}^1\)-spectra \(E=E_{c=0}\) in the Chow heart in an essentially algebraic fashion. The authors provide computational evidence for the power of this approach by addressing the resulting spectral sequence for the motivic sphere spectrum \(\mathbf{1}\) over \(\mathbb{C}\), \(\mathbb{R}\), and finite fields.

Reviewer: Oliver Röndigs (Osnabrück)

MSC:

14F42	Motivic cohomology; motivic homotopy theory
18G80	Derived categories, triangulated categories
55Q45	Stable homotopy of spheres

Keywords:

motivic stable homotopy theory; stable homotopy groups of spheres; Adams-Novikov spectral sequence

Citations:

Zbl 1407.55007; Zbl 1485.55017

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